Hybridized bands and stacking-dependent band edges in ferromagnetic Fe3GeTe2/CrGeTe3 moiré heterobilayer

Owing to unique fundamental physics and device applications, twisted moiré physics in two-dimensional (2D) van der Waals (vdW) layered magnetic materials has recently received particular attention. We investigate magnetic vdW Fe3GeTe2 (FGT)/CrGeTe3 (CGT) moiré heterobilayers with twist angles of 11° and 30° from first-principles. We show that the moiré heterobilayer is a ferromagnetic metal with an n-type CGT layer due to the dominant spin-majority electron transfer from the FGT layer to the CGT layer, regardless of various stacked structures. The spin-majority hybridized bands between Cr and Fe bands crossing the Fermi level are found regardless of stacking. The band alignment of the CGT layer depends on the effective potential difference at the interface. We show that an external electric field perpendicular to the in-plane direction modulates the interface dipole and band edges. Our study reveals a deeper understanding of the effects of stacking, spin alignment, spin transfer, and electrostatic gating on the 2D vdW magnetic metal/semiconductor heterostructure interface.

vdW materials can be prevented due to the vdW interlayer distance, and thus the heterostructure has an abrupt interface. Because of this advantage, the band edges of heterostructures combined with 2D vdW metals and semiconductors have been investigated with and without electrostatic gating for device applications. Several theoretical studies have shown the electrostatic gating effect on the band edges of nonmagnetic 2D vdW metal/ semiconductor heterobilayers 1,5 . However, first-principles studies on the band edges of 2D vdW magnetic twisted moiré metal/semiconductor heterobilayers have rarely been performed 11 .
In this work, we conducted a first-principles study on the structural, electronic, and magnetic properties of the FGT/CGT moiré heterobilayer with twist angles of 11° and 30°. First, we examined the atomic distortion and unfolded band structures in a large moiré supercell with a twist angle of 11°, and then investigated the band edges depending on stacking in a small moiré supercell with a twist angle of 30° with and without electrostatic gating. Each small moiré heterobilayer having one of six possible stacked structures was an FM metal with an n-type CGT layer, induced by the effective spin-majority electron transfer from FGT to CGT. We also compared the band structures of the FM and AFM heterobilayers, having FM and antiferromagnetic (AFM) spin alignments between FGT and CGT. We found that only the FM heterobilayer had the hybridized bands between the Cr and Fe atoms. Further, we explained the stacking-dependent band edges of the CGT layer in terms of the effective potential difference. Finally, we addressed the external electric field effect on the band edges of the CGT layer in the FM heterobilayer.

Methods
We conducted first-principles electronic structure computations using the noncollinear density functional theory (DFT) with the Hubbard repulsion, U, and exchange interaction, J, as well as the spin-orbit coupling (SOC). We used the norm-conserving pseudopotentials 44 with the SIESTA code 45 . We adopted the generalized-gradient approximation, parameterized by the Perdew-Burke-Ernzerhof (PBE) formula. Further, we used the rotationally invariant U approach 46 to describe the electron correlation for Cr 3d orbitals. The atomic structures were optimized using the spin-polarized optB88 vdW method until all atomic forces were less than 0.001 eV Å −1 except for a huge moiré supercell with a twist angle of 11°, where the force criteria were 0.036 eV Å −1 . In addition, we generated a semicore pseudopotential for Cr using the (3s 2 , 3p 6 , 3d 5 , 4s 0 ) valence configuration; the valence configurations for Fe, Ge, and Te pseudopotentials were (3d 7 , 4s 1 , 4p 0 ), (3d 10 , 4s 2 , 4p 2 ), and (4d 10 , 5s 2 , 5p 4 ), respectively. We generated real-space grids with a cutoff energy of 400 Ry and expanded the electronic wave functions using pseudoatomic orbitals (PAOs) of the double-zeta polarization basis set. We utilized a 16 × 16 × 1 k-grid for all structure relaxation and electronic structure calculations except for a large moiré supercell with a twist angle of 11°, for which we applied a 2 × 2 × 1 k-grid. For ML CGT, we obtained U = 5.97 eV and J = 1.00 eV for Cr 3d orbitals with a cutoff radius of 1.58 Å by using a PAO-based constrained DFT (cDFT) method 45,47 . However, our previous study confirmed that the suitable theoretical bandgap close to the experimental one was obtained by the PBE + U + SOC method using a U in the range of 3-4 eV and J = 1 eV 48 . Therefore, we selected U = 3.5 eV and J = 1 eV for our calculations. To remove the interactions between FGT and CGT, we applied a vacuum space of 16 Å.

Results and discussion
Twisted FGT/CGT moiré heterobilayer. Bulk FGT has a layered hexagonal structure with space group P6 3 /mmc and experimental hexagonal cell parameters a = b = 4.030 and 3.991 Å and c = 16.343 and 16.336 Å from refs. 47 and 48, respectively. In ML FGT, each unit cell contains three Fe atoms occupying two nonequivalent Fe positions; Fe1 and Fe2 denote the relevant atoms. ML FGT consists of five sublayers: the first and fifth are occupied by Te atoms, the second and fourth by Fe1 atoms, and the third by Fe2 and Ge atoms. Our relaxed lattice constants for pristine ML FGT that are obtained using the spin-polarized optB88 vdW method are a = b = 4.091 Å. Bulk CGT has a rhombohedral symmetry with space group R3 and experimental hexagonal cell parameters a = b = 6.8196 Å and c = 20.3710 Å at 5 K 27 . ML CGT has two Cr atoms in a unit cell. ML CGT consists of five sublayers: the first and fifth are occupied by Te atoms, the second and fourth by Ge atoms, and the third by Cr atoms. ML CGT has a honeycomb lattice of edge-sharing CrTe 6 octahedrons, and Ge dimers located at the center of the hexagon. Our relaxed lattice constants for pristine ML CGT that are obtained using the spinpolarized optB88 vdW method are a = b = 6.940 Å. The point group for ML FGT and ML CGT is D 3h and C 1 , respectively. Thus the heterobilayer built by ML FGT and ML CGT has the C 1 symmetry.
The heterobilayer of ML FGT and ML CGT has a moiré pattern due to the lattice mismatch. In principle, the twist angle φ between the supercell lattice vectors of the FGT and CGT layers for a moiré supercell (i.e., φ between − → L CGT,1 and − → L FGT,1 in Fig. 1a Fig. 1a. Thus, the moiré wavelength, λ, becomes = 2π 50 . The right panel in Fig. 1a shows λ (red lines) with respect to the twist angle φ between ML CGT and ML FGT optimized using the spin-polarized optB88 vdW method. The moiré superlattice vectors can be expressed as where m, n, p, and q are integers. We can obtain the commensurate moiré lattice vectors by applying a small biaxial strain △ between ML CGT and ML FGT layers. △ is estimated The moiré wavelengths within the constrained condition of △ being less than 1.0% are shown as blue squares in the right panel in Fig. 1a. Here the twist angle φ is determined as cosφ = − → L CGT,1  Fig. 1c,d, respectively. The in-plane displacement vectors indicated by red arrows show more in-plane distortion in CGT than in FGT and are largest at the Te CGT layer. In particular, the in-plane displacement vectors of all Te CGT atoms have consistent patterns regardless of stacking, as depicted by black circles, but those of other atoms vary depending on stacking. The z-coordinates of Te CGT and Te FGT atoms inside the green shaded region in (b) vary depending on stacking, as shown in Fig. 1d. The out-of-plane distortion of the Te CGT layer is much larger than that of the Te FGT layer. The variation of interlayer distance between two interface Te layers is largest in the area where Te CGT atoms are closely located above Te FGT atoms.
The unfolded band structures of the moiré heterobilayer with a twist angle of 11.39° reveal some distinct features compared with the unit-cell band structures of a single layer, as shown in Fig. 1e. All the band structures were obtained using the noncollinear PBE + U + SOC method when U = 3.5 eV and J = 1.0 eV for Cr 3d orbitals, and an FM spin alignment between FGT and CGT was considered. The average magnetic moment is 3.64 µ B / atom for Cr, 2.61 µ B /atom for Fe1, and 1.  www.nature.com/scientificreports/ energy denoted by red arrows. On the other hand, the unfolded band structures of CGT (HB CGT) exhibit a reduced bandgap compared with that of ML CGT, and the conduction bands of HB CGT cross the Fermi level, generating an n-type CGT layer. Also, unexpected bands of HB CGT are observed as indicated by a blue arrow compared to ML CGT. These intriguing electronic features of moiré heterobilayer motivate us to examine the critical elements that influence band structures.
To investigate the electronic structures in terms of stacked structures, we constructed small moiré supercells of the FGT/CGT heterobilayer forming only one stacked structure with a twist angle of 30° and △ = 1.36%, as shown in Fig. 2a. The in-plane supercell indicated by a solid green diamond is equal to the 1 × 1 in-plane unit cell of ML CGT and also corresponds to the 30° rotated √ 3 × √ 3 in-plane supercell of ML FGT, drawn by an open black diamond. Inside the supercell, there are two CrTe 6 octahedrons.
ML FGT is an Ising-type FM metal, whereas ML CGT is an Ising-type FM semiconductor. To predict the magnetic structures of the heterobilayer, we compared the total energy of the heterobilayer with the FM and antiferromagnetic (AFM) spin alignments between ML FGT and ML CGT along the direction perpendicular to the interface. As shown in the left panel in Fig. 2a, we considered 15 different stacked structures while fixing ML FGT and sliding ML CGT. Concretely, we considered six different stacked structures in two diagonal directions of the in-plane supercell and three different stacked structures in the direction of the in-plane lattice. All the atoms were optimized during the relaxation in the x-, y-, and z-directions. The FM spin alignment between ML FGT and ML CGT always has lower energy than the AFM spin alignment at each stacked structure, as shown in the right panel in Fig. 2a.
We categorized the FM energy into three classes: the lowest, E L , intermediate, E I , and the highest energy, E H . The corresponding stacked structures are depicted in the top panel in Fig. 2b. The background lattice shows the FGT top view. The red and purple triangles represent the interface Te triangles in the FGT and CGT layers, respectively. We drew two purple triangles on the bottom Te atoms in CGT related to two CrTe 6 octahedrons inside a supercell. In the E I structure, the interface Te CGT atom is above the Fe1 FGT atom; in the E L structure, it is above the Ge FGT atom; and, in the E H structure, it is located above the interface Te FGT atom. Three more possible   Figure 2c presents the total energy difference with the FM (red) and AFM (blue) spin alignments, as well as the interlayer distance (brown) between the interface Te atoms of the FM structures. All the atoms and axes were optimized during the relaxation in the x-, y-, and z-directions using the spin-polarized optB88 vdW method. The total energy difference changes gradually from E I through E L to E H (left panel) and from E I2 through E I3 to E H2 (right panel). The total energy difference and the interlayer distance are the smallest for E L and the largest for E H and E H2 . E I , E L , E H , E I2 , E I3 , and E H2 structures have interlayer distances of 3.39 Å, 3.22 Å, 3.78 Å, 3.33 Å, 3.31 Å, and 3.71 Å, respectively. At each FM heterobilayer, the lattice constants of the CGT and FGT layers become tensile strained and compressive strained, respectively, as summarized in Table 1. The total energy of FM spin alignment is lower than that of AFM for E I , E L , E H , E I2 , E I3 , and E H2 by 0.02 eV, 0.04 eV, 0.02 eV, 0.02 eV, 0.05 eV, and 0.03 eV, respectively. At the FM spin alignment, the total energy of the E L structure is lower than that of E I , E H , E I2 , E I3 , and E H2 structures by 0.08 eV, 0.30 eV, 0.06 eV, 0.03 eV, and 0.29 eV, respectively. Figure 2d shows the binding energy, E b , per Cr atom along the z-direction regarding the small displacement. At the equilibrium position, the binding energy per Cr atom obtained using the spin-polarized optB88 vdW method has a negative value of − 0.44 eV, − 0.58 eV, and − 0.54 eV for E H , E L , and E I , respectively. The lower the total energy is, the lower the binding energy is. In the PBE + U + SOC method, the FM spin alignment also has lower energy than the AFM spin alignment when U is 3-4 eV and J = 1 eV for all the stacked structures, as shown in Fig. 2e.
The band alignment of FM and AFM heterobilayers was investigated. Figure 3 shows the magnetic moment (left panels), spin-majority (up) and spin-minority (down) band structures (middle panels), and atom projected density of states (PDOS, right panels) of (a) the FM CGT-only system, (b) FM FGT-only system, (c) FM FGT/ CGT heterobilayer for E H , and (d) AFM FGT/CGT heterobilayer for E H in the supercell band lines. The band structures were calculated by using the noncollinear PBE + U + SOC method on Cr 3d orbitals with U = 3.5 eV and J = 1.0 eV. It should be noted that the CGT (FGT)-only system refers to only the CGT (FGT) layer and no FGT (CGT) layer while maintaining the relaxed heterobilayer structures. The band projections onto the spinup and spin-down bands of Cr or Fe atoms are distinguished by red and blue colors, respectively, with various color intensities proportional to the projection magnitude to examine the band alignment. The semiconducting bandgap of 0.43 eV is shown for a Cr atom in the FM CGT-only system (Fig. 3a), while the metallic band structure is shown for a Fe2 atom in the FM FGT-only system (Fig. 3b). The magnetic moment is 3.69 µ B /atom for Cr, 2.62 µ B /atom for Fe1, and 1.67 µ B /atom for Fe2. The atom PDOS of a Fe2 atom in (b) indicates an equivalent magnitude of spin-up and spin-down PDOS near the Fermi level. The CGT-only system features a conduction band minimum (CBM) at a k point between the Ŵ and K points dominated by spin-up empty Cr 3d e g bands. Its valence band maximum (VBM) at the Ŵ point is dominated by Te 5p orbital characteristics, similar to the pristine ML CGT.
On the other hand, the heterobilayer formed by CGT-and FGT-only systems is an FM metal, as shown in Fig. 3c. The bands projected on a Cr atom show the initial Cr bands observed in the CGT-only system as well as unique spin-up hybridized bands with Fe2 bands crossing the Fermi level, as indicated by green and blue arrows, respectively. The hybridization is not related to Fe1 bands, as seen by the band projected on a Fe1 atom. Furthermore, when the hybridized spin-up bands are ignored, the CBM of Cr bands appears to be at the spin-down bands, located at an energy close to the Fermi level, generating an n-type CGT layer. The n-type CGT means that electrons are transferred from the FGT to CGT layers. In the right panel of Fig. 3c, the downward-band shifts of a Cr atom in the heterobilayer can be observed in the atom PDOS. The spin-up Fe2 bands shift higher in energy near the Ŵ point when the system switches from an FGT-only system to a heterobilayer, as indicated by the black arrows in (b) and (c). These upward shifted spin-up Fe2 bands might also originate from the electron transfer from the FGT to CGT layers. Unlike the FM heterobilayer, the AFM-heterobilayer-band structures do not exhibit hybridized bands between FGT and CGT bands but do expose an n-type CGT layer via the spin-up and spin-down CBM of the original Cr bands crossing the Fermi level. AFM heterobilayer, like FM heterobilayer, features shifted spin-down Fe2 bands near the Ŵ point that are higher in energy than those of the FGT-only system (black arrow in (d)), indicating electron transfer from the FGT to CGT layers. Figure 4 shows the spin-up (red) and spin-down (blue) band structures projected on a Cr atom of all the stacked structures with the FM spin alignment. All the band structures exhibit spin-up hybridized bands crossing  Fig. 5. The relative band alignment between the FGT and CGT layers before and after stacking is shown in Fig. 5a. Before stacking, the calculated work function of pristine ML CGT, 4.33 eV, is slightly larger than that of pristine ML FGT, 4.08 eV. After stacking, atomic and electronic rearrangements occur, especially near the interface. The electron transfer from the FGT to CGT layers causes an n-type CGT layer to be formed. When the work function of the metal is less than that of the n-type semiconductor, band bending at the interface generates an Ohmic contact between the metal and the n-type semiconductor. Our vdW heterobilayer, on the other hand, has the vdW interlayer space without any extra bonds between the two constituent layers, resulting in an abrupt interface. There is only the interface dipole at the interface, which is dictated by electron distributions. The interface dipole is defined by the associated effective potential difference at the interface, δV eff . The electrostatic potential difference between the vacuum levels of the FGT and CGT layers in the heterobilayer can be used to determine δV eff , which is reported in Table 2. In addition, δV eff is compared with that calculated using the electron density difference, and the relationship between δV eff and electron transfer is explored in Fig. 6. The band edges of the n-type CGT layer in the heterobilayer were computed to quantify the band alignment.  Table 2. According to Fig. 5c, the variations in band edges and δV eff across different stacked structures are similar, suggesting a significant correlation. For example, the magnitude order in n,↑,↓ is E I > E L > E H , which is the same as in δV eff . In all the stacked structures, p,↑,↓ is larger than n,↓ , confirming the n-type CGT, and the biggest variation in band edges across different structures is roughly 0.1 eV.
To study the spin/electron transfer and δV eff derived from the electron density difference, the spin-resolved valence electron density difference δρ ↑,↓ (z) = ρ HB↑,↓ (z) − ρ FGT↑,↓ (z) − ρ CGT↑,↓ (z) was computed. Here,  Fig. 6b) illustrates that the spin-up density transfers from the FGT to CGT layers, and the spin-down density transfers from the CGT to FGT layers, resulting in the effective electron density difference (black dotted lines,δρ eff (z) = δρ ↑ (z) + δρ ↓ (z) ). A considerable electron transfer from Fe2 atoms to Cr atoms is observed in the atomic-structure figure via the bonding between Fe2 atoms and interface Te FGT atoms, via the vdW spacing between interface Te FGT atoms and interface Te CGT atoms, and via the bonding between interface Te CGT atoms and Cr atoms. Whereas the electron transfer from Fe1 atoms to Cr atoms is absent due to no direct bonding between Fe1 atoms and interface Te FGT atoms. This electron-transfer pathway explains the hybridized band between Cr atoms and Fe2 atoms. In the interface region, the interface dipole appears as electron depletion and accumulation, represented by + and − signs, respectively, resulting in an effective electric field and a potential difference at the interface. Specifically, the one-dimensional Gauss law can be used to compute the effective electric field, E eff (z) , and effective potential difference, δV eff (z) , at the interface, as illustrated in the middle panel of Fig. 6b. Q in = z z0 δρ eff (z)dz gives the electron density inside the gray dashed box in the top panel, where z 0 is an arbitrary point z at the left side vacuum outside the FGT layer. Because the electric field at the left side vacuum is zero, E eff (z) = Q in (z)(ε r ε o ) −1 gives the electric field at location z, where ε o is the vacuum permittivity, and ε r is the position-dependent relative permittivity. Since ε r is around 1.0, 13.50, and ∞ in the vacuum, CGT, and FGT layers, respectively, the effective electric field exists only at the interface between the two layers. Finally, δV eff (z) for electrons at the interface may be calculated using δV eff (z) = − z 0 E eff (z)dz ; further, it is verified that δV eff (z) equals the sum of δV eff ,↑ (z) and δV eff ,↓ (z) . The Gauss law yields δV eff values (V o in Fig. 6b) of 0.28 eV, 0.32 eV, and 0.37 eV for E H , E L , and E I , respectively, which are extremely similar to the electrostatic potential differences of 0.28 eV, 0.31 eV, and 0.35 eV, respectively. Notably, the general signs for E eff and δV eff defined for a positive charge are the inverse of those defined for the electron in this work. www.nature.com/scientificreports/ The spin variation ( �ρ ↑,↓ = ρ HB,↑,↓ − ρ 1L,↑,↓ ) and the charge variation ( �ρ = �ρ ↑ + �ρ ↓ ) per single layer were quantitatively investigated using the spin-dependent Mulliken electron differences at the atomic sites, as shown in the bottom panel of Fig. 6b. The spin-up electron in the FGT layer moves to the spin-up electron in the CGT layer by 0.10 e. In contrast, the spin-down electron in the CGT layer moves to the spin-down electron in the FGT layer by 0.05 e, resulting in a 0.05 e effective spin-up electron transfer from FGT to CGT. Also, the magnetic moment variation per single layer, i.e., �ρ ↑ − �ρ ↓ is m FGT = − 0.15 µ B and m CGT = 0.15 µ B . In addition, the magnetic moment variation by the atomic site ( δm , solid black square) demonstrates that magnetic moments are slightly larger at Cr sites and slightly smaller at Fe2 sites than those of a single layer. The spin-down variation (blue open circle) contributes more to the magnetic moment changes at the Cr and Fe2 sites than the spin-up variation (red open circle).   Fig. 6a, is reversed compared to that in the FM heterobilayer due to the FGT layer's reversed spin-up and spin-down bands in the AFM heterobilayer. However, the electron distribution at the interface in the AFM heterobilayer is identical to that in the FM heterobilayer, resulting in similar E eff and δV eff . As demonstrated in the atomic-structure figure in Fig. 6a, the practical electron-transfer pathway in the AFM heterobilayer is comparable to that in the FM heterobilayer. According to the quantitative spin variation per single layer, complex spin transfer between FGT and CGT results in a 0.05 e effective electron transfer from FGT to CGT. Also, the magnetic moment variation per single layer is m FGT = 0.11 µ B and m CGT = 0.07 µ B . The magnetic moments of both the Cr and Fe2 sites are slightly larger than those of a single layer.
Electrostatic gating effect. This study considers positive and negative electric fields within 0.5 VÅ −1 , perpendicular to the in-plane direction. When utilizing the PBE + U + SOC method with U = 3.5 eV and J = 1.0 eV to compare the total energies of the FM and AFM E H , E I , and E L structures concerning the external electric field, FM structures are always more stable than AFM structures, as shown in Fig. 7a. Figure 6c,d show the spin/electron transfer for the FM E H structure. We show only E H results because the main effect of electrostatic gating on all stacked structures is almost comparable. Our results show that the external electric field mainly modulates the spin-majority spin transfer between FGT and CGT. As shown in Fig. 6c, the variation of δρ ↑,↓ increases when a negative electric field of − 0.5 VÅ −1 is introduced. Quantitatively, the spin-up electron moves from FGT to CGT by 0.21 e, whereas the spin-down electron moves from CGT to FGT by 0.03 e, resulting in a 0.17 e effective spin-up electron transfer from FGT to CGT. As a result, the magnetic moment variation per single layer increases, with m FGT = − 0.25 µ B and m CGT = 0.23 µ B , respectively. More enhanced δm at Cr sites and more reduced δm at Fe2 sites are also induced by the negative electric field. In the interface region, the negative electric field promotes electron depletion (++) and accumulation (−−), resulting in larger E eff and δV eff (V o + V a , where V a = 0.76 eV in Fig. 6c).
In contrast, the variation of δρ ↑,↓ decreases under a positive electric field of 0.5 VÅ −1 , as shown in Fig. 6d Consequently, the spin-up electron transfer is relatively tiny, whereas the spin-down electron transfer from CGT to FGT is 0.06 e, resulting in a 0.06 e effective spin-down electron transfer from the CGT to FGT layers. This spin transfer results in less increased and decreased δm at Cr and Fe2 sites, respectively, and reduces the magnetic The FGT/CGT heterobilayer is analogous to a Schottky diode in device performance, except for the opposite electron distribution in the interface. In a Schottky diode depletion zone, electron accumulation occurs near the metal, and electron depletion occurs near the n-type semiconductor. However, the electron distribution in our heterobilayer is reversed. A negative electric field to the heterobilayer, i.e., reverse bias, is achieved by connecting the battery's positive terminal to CGT and the negative terminal to FGT. The positive terminal to CGT allows FGT's electrons to migrate more to CGT, resulting in a stronger interface dipole (middle panel in Fig. 6c) and CGT's lower energy-band shift relative to the Fermi level. Furthermore, as illustrated in the left panel of Fig. 7b, the CGT's lower energy-band shift defines narrower n-type band edges but wider p-type band edges.
In contrast, a positive electric field, i.e., forward bias, is achieved by connecting the battery positive terminal to FGT and the negative terminal to CGT. The positive terminal to FGT causes CGT's electrons to move to FGT, resulting in a smaller or inverted interface dipole (middle panel in Fig. 6d). On the other hand, the negative terminal to CGT leads to the CGT's higher energy-band shift relative to the Fermi level. Furthermore, as shown in the right panel of Fig. 7b, the CGT's higher energy-band shift sets wider n-type band edges but narrower p-type band edges. Figure 7c illustrates how the band edges change with the external electric field. As the external electric field increases, n-type band edges widen, and p-type band edges narrow, regardless of the stacked structures of E H , E I , and E L . However, slightly different band-edge values depending on the stacked structures cause differences in the contact type as the external electric field changes. In the E L heterobilayer, the n-type contact is maintained in the range of − 0.5 VÅ −1 to 0.5 VÅ −1 . In the E H heterobilayer, the n-type contact changes to a particular contact ( � n,↓ < 0 ) below − 0.3 VÅ −1 . In the E I heterobilayer, the n-type contact changes to p-type contact above 0.36 VÅ −1 . These differences, depending on the different stacked structures, originate from the different initial potential differences ( δV eff when there is no external electric field) due to different asymmetric interface structures (Supplementary Information).

Conclusions
We presented the structural, electronic, and magnetic properties of layered FGT/CGT twisted moiré heterobilayers and the electrostatic gating effect. Our results show that the moiré heterobilayer with a twist angle of 30° is an FM metal with an n-type CGT layer regardless of various stacked structures. In the spin transfer of the FM heterobilayer, an effective spin-up electron transfer occurs from FGT to CGT. Furthermore, the spin-up hybridized bands between Cr and Fe2 atoms crossing the Fermi level are found only in the FM heterobilayer, not in the AFM heterobilayer, independent of stacking. It also turns out that the band alignment of the CGT layer is explained by the effective potential difference at the interface. Our findings further reveal that, in the FM heterobilayer, the external electric field regulates the spin-up electron transfer between the FGT and CGT layers and affects the band edges of the CGT layer. Our study could provide helpful information for understanding the effects of stacking, spin alignment, spin transfer, and electrostatic gating in the magnetic 2D vdW metal/ semiconductor heterobilayer.